Image Processing Techniques in Emission Tomography - Correction Techniques in Emission Tomography - page 123

Wiener filtering
The Wiener filter was introduced in the 1940s by Norbert Wiener. It is an
optimal filter in the sense of minimizing the mean-squared error (MSE) of the
denoised image I d and the true image I u . In the following, a local adaptive
version of the Wiener filter, as described in [39], is discussed. For each pixel
(x;y), local statistical characteristics such as the mean (x;y) and the variance
2 (x;y) are estimated from a local neighborhood and are used to calculate
the new pixel value I d (x;y):

f (x;y)
f (x;y) + avg

I d (x;y) = (I(x;y) (x;y)) + (x;y) ; (7.5)

where avg denotes the average of all local variances (

) and serves as a
variance estimation of the additive noise N of I. Further f (x;y) is dened
as

· ; ·

( 2 (x;y) avg ; if 2 (x;y) > avg
0

f (x;y) = (7.6)

otherwise :

In Figure 7.1(d) the effect of the adaptive character of the local Wiener filter
can be clearly seen in the area around the heart. The noise in the background
is filtered strongly whereas the region of the heart with high uptake is filtered
to a lesser extent.

7.2.2 Fourier transform domain
Contrary to the image domain, images can be represented by their inherent
frequencies in the Fourier transform domain. As noise is primarily present in
high frequencies this is a suitable representation to eliminate noise eciently.
In Fourier domain based filtering the noisy image I is transformed to
frequency space using F, the (fast) Fourier transform (FFT). A filter mask
M F is developed in frequency space and applied to the transformed image
F(I). Subsequently, the filtered frequency image is transformed back to image
space using the inverse FFT F 1 :

I d = F 1 (M F · F(I)) ; (7.7)

where the multiplication is applied point-wise. As F 1 can lead to complex
values, the real part of the inverse FFT should be taken here. Obviously, the
filtering or denoising result depends on the choice of an adequate filter mask
M F .
We depict the proceeding of Fourier domain filtering with a frequency-
based realization of the mean filter, as illustrated in Figure 7.2. Equation (7.3)
together with the convolution theorem (C.T.) leads to

I d = M ? I = F 1 (F(M ? I)) C:T = F 1 (F(M)

· F(I)) ; (7.8)